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Theorem psubclssatN 38407
Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclssat.a 𝐴 = (Atomsβ€˜πΎ)
psubclssat.c 𝐢 = (PSubClβ€˜πΎ)
Assertion
Ref Expression
psubclssatN ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† 𝐴)

Proof of Theorem psubclssatN
StepHypRef Expression
1 psubclssat.a . . 3 𝐴 = (Atomsβ€˜πΎ)
2 eqid 2737 . . 3 (βŠ₯π‘ƒβ€˜πΎ) = (βŠ₯π‘ƒβ€˜πΎ)
3 psubclssat.c . . 3 𝐢 = (PSubClβ€˜πΎ)
41, 2, 3psubcliN 38404 . 2 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ (𝑋 βŠ† 𝐴 ∧ ((βŠ₯π‘ƒβ€˜πΎ)β€˜((βŠ₯π‘ƒβ€˜πΎ)β€˜π‘‹)) = 𝑋))
54simpld 496 1 ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 βŠ† 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  β€˜cfv 6497  Atomscatm 37728  βŠ₯𝑃cpolN 38368  PSubClcpscN 38400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-psubclN 38401
This theorem is referenced by:  pmapidclN  38408  psubclinN  38414  paddatclN  38415  pclfinclN  38416  poml6N  38421  osumcllem3N  38424  osumcllem9N  38430  osumcllem11N  38432  osumclN  38433
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